Algebraic Geometry Seminar: Alexander Woo

We characterize Schubert varieties (for GLn) which are local complete intersections (lci) by the combinatorial notion of pattern avoidance. For the Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighborhoods at the identity. Although the statement only requires ordinary pattern avoidance, showing the other Schubert varieties are not lci appears to require more complicated combinatorial ideas which have their own geometric underpinnings. The Schubert varieties defined by inclusions, originally introduced by Reiner and Gasharov, turn out to be an important subclass of lci Schubert varieties. Using the explicit equations at the identity for the lci Schubert varieties, we can recover formulas for some of their local singularity invariants at the identity as well as explicit presentations for their cohomology rings.